We prove that the Julia set of a quadratic polynomial which admits an infinite sequence of unbranched, simple renormalizations with complex bounds is locally connected. The method in this study is three-dimensional puzzles.

J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J. -C. Yoccoz, Topological Methods in Modern Mathematics, A Symposium in Honor of John Milnor's Sixtieth Birthday (1993). MR 94c:58172

Yunping JiangAffiliation:
Department of Mathematics, CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016 and Department of Mathematics, Queens College of CUNY, Flushing, New York 11367
Email:
yunqc@jiang.math.qc.edu